The
Budyko hypothesis (BH) is an effective approach to investigating long-term
water balance at large basin scale under steady state. The assumption of
steady state prevents applications of the BH to basins, which is unclosed, or
with significant variations in root zone water storage, i.e., under unsteady
state, such as in extremely arid regions. In this study, we choose the Heihe
River basin (HRB) in China, an extremely arid inland basin, as the study
area. We firstly use a calibrated and then validated monthly water balance
model, i.e., the
The Budyko hypothesis (hereafter BH) was postulated by a Russian climatologist, Mikhail Ivanovich Budyko, to analyze regional differences in long-term annual water and energy balance (Budyko, 1948). The BH's mean annual water balance is described by the evapotranspiration ratio and the climate aridity index. The BH becomes an effective approach to investigating the influence of climate change on mean annual runoff and evapotranspiration (Donohue et al., 2011; Xiong et al., 2014). There are various equations to describe the BH. Some empirical equations without parameters were proposed by Schreiber (1904), Ol'dekop (1911), Budyko (1948) and Pike (1964) (see Table 1). These equations explicitly include climate variations (radiation, precipitation, evapotranspiration and air temperature) and do not deal with recently recognized important catchment properties, such as characteristics of groundwater system, vadose zone properties, vegetation. Hence, attempts have been made to introduce physical parameters in these empirical equations (Mezentsev, 1955; Fu, 1981; Milly, 1993; Zhang et al., 2001; Yang et al., 2007, 2008). These physical parameters are a collection of myriad catchment characteristics (topography, vegetation, soil, and groundwater, etc.) and are therefore difficult to measure (Gerrits et al., 2009). These equations with a single parameter, however, provide the flexibility of using the BH over long-term timescales.
The BH assumes steady-state conditions. Firstly, the studied basin must be natural and closed, which means that the local precipitation is the only water source to the evapotranspiration. Recently, the BH has been widely used to investigate the interannual variability of precipitation partitioning (Gerrits et al., 2009), separation of runoff trends (H.-Y. Li et al., 2014; Xiong et al., 2015), evapotranspiration change (Savenije, 1997) and water storage change (Istanbulluoglu et al., 2012; Gao et al., 2014). These studies show that hydrological processes have been greatly affected by the climate change and intensive change of land cover owing to human activities. These human activities such as urbanization, withdrawing groundwater, hydraulic engineering, deforestation etc. are significantly changing natural hydrological cycle and breaking the original water balance to form a new balance under the new hydroclimatic conditions. For example, the transferring water becomes the new water source of the basin to evapotranspiration due to the implemented inter-basin water transfer project (Bonacci and Andric, 2010). In dry regions, croplands expanded with irrigation, which increased water availability for evapotranspiration (Gordon et al., 2005). Land use/cover changes have also caused the change of runoff (J. Li et al., 2014). Nowadays, most of the inhabited basins have been developed or disturbed by large-scale human actives. Therefore, lots of basins were no longer closed or natural, and the relationship between annual evapotranspiration ratio and potential evapotranspiration ratio hardly meets the first condition of the BH, which presents a great challenge in applying the BH in unclosed basins.
Secondly, water storage change can be assumed to be negligible at the basin scale and at long-term timescale. However, over finer temporal scales, it becomes increasingly concerned of the importance of water storage in water balance in the Budyko framework. For example, Wang et al. (2009) found that the inter-annual water storage change should be considered due to the hysteresis response of the base flow to the inter-annual precipitation change in Nebraka Sand Hills. Zhang et al. (2008) considered the impacts of soil water and groundwater storage and developed a monthly water balance model based on the BH with application in 265 catchments in Australia. Yokoo et al. (2008) highlighted the importance of soil water storage change in determining both annual and seasonal water balances. Wang (2012) evaluated changes in inter-annual water storage at 12 watersheds in Illinois using the field observation of long-term groundwater and soil water and found that the impact of inter-annual water storage changes on the water supply in the BH need to be considered. Chen et al. (2013) defined the difference between rainfall and storage change as effective precipitation to develop a seasonal model for construction long-term evapotranspiration. Therefore, water storage change should be taken into account as the important part of the steady-state assumption of the BH (Zhang et al., 2008).
Different Budyko equations for the mean annual water-energy balance.
Note:
In summary, it has been more and more recognized that water systems are no
longer natural to different extents (Sivapalan et al., 2011). Hence, it
presents a great challenge to apply the BH to unsteady-state conditions
(unclosed basins or intense water storage changes). The BH has been widely
applied to mild arid basins with precipitation of 300–400 mm and aridity
index of less than, for example, 5, such as over northern China (Yang et al.,
2007), the southwestern regions of MOPEX catchments (Gentine et al., 2012;
Carmona et al., 2014) and the west of Australia (Zhang et al., 2008).
However, it is rare to apply the BH in extremely arid environments (say, the
aridity index over 5), where water systems are typically unclosed with
intense human interference and irrigation. For example, rivers in the arid
region of northwestern China are typically from upper mountains with little
human interference, and flow through middle regions with intensive irrigation
and human interferences and finally into extremely dry desert plains. To
investigate it in more detail, we choose the Heihe River basin (HRB), the
second largest arid inland basin in northwestern China (mean annual aridity
index
In the original BH, the basin is a natural hydrologic unit, and the only
possible water source for evapotranspiration is the local precipitation. The
annual or monthly water balance equation can be written as
Because of human interferences (land cover change, dams, irrigation and
other withdrawals) to the hydrologic system worldwide, the water supply to
evapotranspiration in a basin has changed. Local groundwater and root zone
water and external water transfer also become new possible water sources.
However, that new non-ignorable part of available water for
evapotranspiration has yet been explicitly considered in the Budyko
framework in an unclosed basin. More specifically, the inflow or/and
inter-basin water transfer may affect the available water for
evapotranspiration largely. By considering that, here we rearrange Eq. (1)
as
As discussed above, in the original Budyko framework, the water supply to
land evapotranspiration is mean annual precipitation, and the energy supply
to land evapotranspiration is estimated by mean annual potential
evapotranspiration. The general Budyko equation can be written as
Water balance analysis in Sect. 2.1 concludes that the water supply in the BH
under the unsteady-state condition is the equivalent precipitation instead of
the local precipitation. So the annual (or monthly) evapotranspiration ratio
is redefined as the ratio of annual (or monthly) evapotranspiration and
equivalent precipitation, and the annual (or monthly) aridity index is
redefined as the ratio of annual (or monthly) potential evapotranspiration
and equivalent precipitation. They are described as follows:
A schematic diagram of the BH under the unsteady-state condition.
The Budyko curves in Eq. (8) with different combinations of
parameters
Under the unsteady-state conditions for a region, when the local
precipitation in the origin Fu's equation is zero, evapotranspiration may not
be zero due to other water sources (e.g., inter-basin water transfer), so
following the derivation of Fu (1981). Equation (4) can be rewritten as
Regional evapotranspiration and soil water cannot be measured directly and
they are usually provided by monthly water balance models. Monthly water
balance models were first developed in the 1940s. From that, many models have
been developed in hydrological studies, such as the
Among these monthly models, the
The partitioning of monthly precipitation
Location of study area and the distribution of hydrological stations and meteorological stations.
The HRB, originating from Qilian Mountains, is the second largest inland
river basin in the arid area of the northwestern China (Fig. 3). The drainage
map and the basin border are extracted using a 90 m resolution digital
elevation model (DEM) data from the Shuttle Radar Topography Mission (SRTM)
website of NASA (
The HRB is divided into six sub-basins according to basin characteristics,
distributed along the eastern and western tributaries, shown in Fig. 3.
Regions I and II are upper mountainous regions with the elevation of
3000–5500 m and belong to the cold and semiarid mountainous zone dominated
by shrubs and trees with mean annual temperature of less than 2
Time series of observed and simulated monthly streamflow using the
The required data for Eq. (8) and the
The daily precipitation data of all stations during 1978–2012 are obtained
from the year book hydrology of China including 28 rainfall stations and the
China Administration of Meteorology including 19 meteorological stations
(Fig. 3). The monthly precipitation of each station is calculated by summing
daily precipitation. The gridded data set with 1 km resolution across the
whole basin is obtained by interpolation of the site data. The monthly
precipitation of the six sub-basins is obtained by the extraction from the
monthly precipitation in the whole basin. Daily meteorological data of 19
stations during 1978–2012 are also available. Daily potential
evapotranspiration is estimated in each station using the FAO
Penman–Monteith equation recommended by Allen et al. (1998). The monthly
ET
The red points in Fig. 3 are the locations of hydrological stations. For the
two upper streams, Gauge S1 controls region I and Gauge S2 controls region
II. For the two middle streams, Gauges S1 and S3 control region III and
Gauges S3 and S4 control region IV. For the two lower
streams, regions V and VI without any runoff flowing out, Gauges S2 and S4
control their inflow, respectively (Fig. 3). Monthly runoff data are obtained
from the year book of hydrology of China and are intended for calibrating the
The natural runoff in regions III and IV were strongly disturbed by human
activities and there is no runoff for regions V and VI and the whole basin.
To validate the outputs of the
In extremely dry basins like the HRB, the lack of observed hydro-climatic
data presents great challenge. A monthly water balance model becomes an
effective tool to estimate actual evapotranspiration, change in soil water
storage and change in groundwater storage. This study employs the
Figure 4 shows the results of the modeled streamflow at monthly timescales in regions I and II. Regions I and II are the water source area of the whole basin with little interference of human activities and both keep relatively natural steady state. The NSE for regions I and II is for 0.92 and 0.83, respectively. The results illustrate that the simulated monthly streamflow agrees well with the observation and other modeled components can be reasonable estimates, for instance, monthly actual evapotranspiration, soil water storage change and groundwater storage change in the two sub-basins.
The mean annual water balance of all regions.
“–” means no runoff; PWS represents the proportion of the root zone water storage change in the total precipitation.
The outputs from the
Over the middle sub-basins (regions III, IV and also V), large areas of
artificial oasis (cropland) is distributed and the streamflow water
intensely disturbed by hydraulic engineering. Hence it becomes almost
impossible to validate the
Comparison between ET simulated by the
Variation of annual water balance for all the regions simulated
using the
To test the “steady-state” assumption of the Budyko framework, it is vital
to examine whether changes in mean annual soil water storage in water balance
approach zero. By using the monthly runoff, evapotranspiration, soil water
and groundwater storage change from the
Variation of average monthly water balance for all regions using the
The fitting parameters of Fu's equation at annual scales.
Comparison of the original Budyko curves (left panel) and the new
Fu-type Budyko curves (middle panel, with
Because this study focuses on the application of the BH at the annual and monthly timescales, the annual and monthly water balance analysis is very critical to understanding the role of water storage and water source change in the BH. Figure 6 describes the variation of annual water balance for the six sub-basins and the whole basin. The most obvious in Fig. 6 is that the proportion of soil water storage change in annual water balance is small compared with the annual precipitation. So the impact of soil water storage change on annual water balance is insignificant and can be also neglected. Moreover, annual evapotranspiration is higher than annual precipitation in regions III–VI and approaches annual precipitation over the whole basin. For water-limited regions, when inflow from other regions is available, the actual evapotranspiration increases with the increased water supply so that the actual evapotranspiration is more than the local precipitation. For the whole basin of the inland HRB, there is no water transferring with other basins, so the evapotranspiration almost approaches the precipitation at the annual timescale due to little variations in the soil water storage changes. In conclusion, the facts that soil water storage change in all basins can be ignored in annual water balance meet the second assumption of the BH, and the results that the annual water balance in regions III–VI and the whole basin have been disturbed do not meet the first assumption of the BH. Therefore, except for regions I and II, the original BH cannot be directly used for those sub-basins and the whole basin.
The fitting parameters of the improved Budyko equation at the monthly scales.
Comparison of the original Budyko curves (left panel) and the new
Fu-type Budyko curves (middle panel, with
Different from the annual timescale, the impacts of monthly changes of soil water storage and groundwater storage behave differently (Fig. 7). The variations of monthly groundwater storage change for all regions are similar to those of runoff (Fig. 7). For those regions with no runoff (regions V and VI and the whole basin), the modeled groundwater storage change is almost zero. This means that the groundwater storage can hardly contribute to the evapotranspiration while the variation of soil water storage is tightly coupled with the evapotranspiration (Fig. 7). For regions I and II and during the winter season, the evapotranspiration is more than the precipitation; the extra water source required by the evapotranspiration is from root zone water storage. After the summer season, the precipitation sharply decreases, but the evapotranspiration slowly decreases by consuming the root zone water storage recharged during the summer season. For regions III–VI, the water supply is more complicated by the interference of monthly inflow water, and the monthly variations of root zone water storage. As shown in Fig. 7, it can be concluded that both the soil water storage change and inflow water have obvious effect on the monthly water balance, whilst the impact of monthly groundwater storage change is negligible.
In summary, due to the complications of the water transfer and soil water storage change, the two assumption conditions for applying the original BH are difficult to meet for the sub-basins and the whole HRB on the monthly timescales, which in turn requires new treatments in the BH as further investigated in the following sections.
Figure 8 (left panel) plots the original Budyko curves for the six sub-basins
and the whole basin. For regions I and II, the points of annual
evapotranspiration ratio and aridity index fall in the domain of water and
energy limit boundary and they can be well fitted by Fu's equation. The
relationship between water and energy in regions I and II can be described by
the original BH as expected in the section above. However, the points of
evapotranspiration ratio and aridity index for other regions exceed the water
limit boundary. And the results show the relationship of water and energy in
regions III–VI, and the whole basin is inconsistent with the original BH.
After using the equivalent precipitation instead of the local precipitation,
the new Fu-type Budyko curves (Eq. (8) with
In summary, if a basin (sub-basin) is closed, the original BH can be applicable at the annual timescale. However under unsteady state, the new Fu-type BH, instead of the original BH, is more applicable to describe the annual water balance.
Again as expected based on the monthly water balance analysis, the points of
monthly evapotranspiration ratio and aridity index exceed the water limit
boundary for all the basins (Fig. 9, left panel). The value of
evapotranspiration ratio can be up to 40, which means that the local
precipitation in original water balance is well below the actual water supply
to the evapotranspiration. The new Fu-type Budyko curves at the monthly
timescale are shown in Fig. 9 in the middle panel (Eq. 8 with setting
Different presentations of the annual water balance for
Five presentations of monthly water balance for region III considering different combinations in the water supply to evapotranspiration.
The fitted values of the parameters in the Budyko curves for regions I to VI
are listed in the Table 4. These curves and the parameters have significantly
seasonal characteristics. For example, the Budyko curves in regions I and II
can be divided to five groups (Fig. 9). The values of the integrated
parameter
In this study, we intended to extend the BH to the annual and sub-annual time scales by explicitly considering the root zone water storage and new water source from other regions. To further investigate it, we choose regions I and III as typical cases in Fig. 10. In region I, as there is no inflow into the region, we can separate the impact of soil water storage and groundwater storage on the BH (Fig. 10a). With subtle differences, the impacts of changes in root zone water storage and groundwater storage on water balance can be almost ignored at annual scales. Region III is another extreme case where only if the role of the inflow water is considered, can the BH perform well under unsteady state (Fig. 10b).
In Fig. 11, we further adopted the approach presented by Chen et al. (2013) to examine the impacts of soil water storage, groundwater storage and inflow water on monthly water balance. We test different combinations in monthly water balance in region III, a midstream sub-basin of the HRB (Fig. 11a–c), and found that when the equivalent precipitation includes the root zone water storage change the BH performs well at the monthly scale. However, the inclusion of the groundwater storage change into the equivalent precipitation does not improve as much (Fig. 11b, c). By examining the impact of monthly inflow water on the BH in region III (Fig. 11d, f), we find that inflow water at the monthly scale has as much impact as that at annual scale. The results presented above highlight the fact that the water supply cannot be the local precipitation only, but should have included root zone water storage change and inflow water.
The Budyko hypothesis (BH) is a useful approach to depicting and understanding the long-term mean water balance at a large basin scale under a steady-state condition. However, river systems worldwide have in fact been disturbed by human activities to different extents. That is important for extremely arid environments (say, the aridity index over 5), especially in China, where water systems are typically unclosed with intense human inference and irrigation. That presents a great challenge if one is applying the BH to those regions under unsteady state, e.g., unclosed or significant variation in soil water storage, or those timescales finer than a year.
To investigate it, we choose an extremely arid inland basin, the Heihe River
basin in China, as the study area, which is divided into six sub-basins based
on catchment hydrologic characteristics. We first calibrate and validate a
widely used monthly water balance model, i.e., the
With the recognition that the inflow water from other regions and the water
storage change are both new possible water sources to evapotranspiration in
unclosed basins, we define the equivalent precipitation (
For an unclosed basin or region, the water supply to evapotranspiration is
defined as equivalent precipitation (
With
For the other constraint,
This research was supported by the National Natural Science Foundation of China (41271049), the Chinese Academy of Sciences (CAS) Pioneer Hundred Talents Program, and an open research fund for the State Key Laboratory of Desert and Oasis Ecology, Xinjiang, Institute of Ecology and Geography, Chinese Academy of Sciences. The authors would like to thank Hubert H. G. Savenige and Wang Ping for their suggestions and support in this research. We are particularly grateful to the two reviewers Wang Dingbao and Gao Hongkai for their efforts and helpful comments.Edited by: H. Li